# Finding flux

## Homework Statement

Find the flux of the vector field through the surface of the closed cylinder of radius c and height c, centered on the z-axis with base on the xy-plane.

## The Attempt at a Solution

Can I just use the divergence theorem here? Find the divergence of the vector field and then multiply it by the volume of the cylinder?

Related Calculus and Beyond Homework Help News on Phys.org
HallsofIvy
Homework Helper
Yes, you can use the divergence theorem but the is not "find the divergence of the vector field and then multiply it by the volume of the cylinder" unless the divergence is a constant. By the divergence theorem, the integral over the surface is the integral of the divergence over the cylinder.

so if it's not a constant then I should integrate it right

HallsofIvy
Homework Helper
Yes!

if I have:

$$\vec{F} = 6x\vec{i} + 6y\vec{j}$$

then according to greens theorem this is 0 right? so therefore the flux is 0 as well?

bump!

dx
Homework Helper
Gold Member
No, it's not zero. You don't use green's theorem in this problem. The theorem you have to use is Gauss' divergence theorem. First find the divergence of F. What is the divergence of (6x, 6y)?

it's just 12... and then multiply it by the volume right, which is $$\pi c^3$$ so it's $$12\pi c^3$$

dx
Homework Helper
Gold Member
That's right. Now just use the divergence theorem. The flux out of the cylinder is ∫(div F)dV over the volume of the cylinder.

Last edited by a moderator:
dx
Homework Helper
Gold Member
What limit?

read my attachment above.. I don't understand the question it self

dx
Homework Helper
Gold Member

so the limit and the flux density are both 12?

dx
Homework Helper
Gold Member
If by 'flux density' you mean 'divergence', then yes.

yea flux density is divergence